The Game Theory Behind Stock Based Comp

If you have worked in a large company that issues stock to its employees, you might have observed that employees don't necessarily feel like "owners". As an employee, it is common to treat the stock price as some externality that is independent of your work.

It is paradoxical considering that the entire premise of stock based compensation is that employees do have an impact on the company's valuation and should be incentivized to increase it by compensating them with the company's shares.

In this post, I want to assume a simple model and illustrate the game theory behind this phenomenon.

Let's say a company called ZikZok has nn employees, and each employee is granted 1 share of ZikZok's stock - meaning the company is 100% owned by its nn employees.

Now, each employee produces vv units of value for the company at cc units of cost, meaning the employee endures cc units of cost for producing vv units of value (this represents the time away from other activities, difficulty of the task, opportunity cost etc...). For their effort, they are compensated with a wage ww (in addition to their one share).

Adding vv units of value to the company makes it proportionately more valuable and the stock price rises by vn\frac{v}{n} (if market cap rises by vv and there are nn shares in circulation, the stock rises by v/nv/n).

Now assume the employee's wage is ww. If an employee works, they produce vv units of value and their payoff is: w+vnc+kvnw + \frac{v}{n} - c + \frac{kv}{n}, where kk is the other number of employees that are producing value (i.e. working).

If the employee doesn't work, they are still getting paid, and are still capturing the value of the kk other workers but does not endure cost cc. Their payoff is: w+kvnw + \frac{kv}{n}.

The employee has an incentive to work if:

payoff(work)payoff(slacking off)w+vnc+kvnw+kvnvnc\begin{aligned}& \text{payoff(work)} \geq \text{payoff(slacking off)} \\& \Leftrightarrow w + \frac{v}{n} - c + \frac{kv}{n} \geq w + \frac{kv}{n} \\& \Leftrightarrow \frac{v}{n} \geq c\end{aligned}

As we can see, the incentive does not depend on the employee's wage nor the number of other employees working, but rather whether the employee is able to capture a fraction of their value added, that is higher than the cost they endure.

The problem arises when nn increases: the fraction vn\frac{v}{n} decreases, but the cost endured by the employee cc remains the same!

There is a point where the employee no longer has an incentive to work, which produces undesirable outcomes for the company.

what's the critical number of employees, such that each of them has an incentive to play?

Let's call ll the employee's leverage, such that v=lcv=lc. If l1l \geq 1, the employee can create more value than the cost they endure. Moreover, for the employee to have an incentive to work (recall vnc\frac{v}{n}\geq c), the number of employees nn should not exceed ll.

If l<1l < 1, the employee's conversion of work to value added is inefficient and there is no number of employees such that they have an incentive to work.

what's a better incentive scheme?

The key insight from this model is that the employee's payoff should not decrease as the number of employees increases.

Instead of allocating 1 share per employee, let's imagine we allocate back a fraction f[0,1]f\in [0,1] of the value produced vv, such that the payoff for working is now w+fvcw + fv - c and the payoff for not working: ww.

We can see that ff should be chosen such that fvcfv \geq c.

This is the thinking behind commission based schemes such as sales teams or traders, which own a percentage of their profit and loss.

Improving the model

There are some adjustments we can make to make the model more realistic:

  • we have considered the cost of working cc to be constant. But, I probably won't accept an offer that I would have accepted as a new grad. For instance, if I am especially productive, I would probably have alternative offers. Moreover, if my basic needs are taken care of, the cost of spending my time working as opposed to playing with my children should grow as well. To add more realism, we can make cc vary (non-linearly) with the wage ww and value produced vv.
  • we have treated individual workers and their value produced as completely independent. However, cooperation might increase value created at no added cost to the employee. There are a number of dynamics that can impact an individual's cost of working cc or value produced vv.

Closing thoughts

When I started undertaking this exercise, I expected to observe something similar to the prisoner's dilemma with nn players. In that setting, almost everybody needs to play along in order to attain the maximal payoff, which becomes exponentially unlikely with nn. What happens is called a coordination failure, where the probability that each plays along should be unreasonably high, thus resulting in a Nash equilibrium whose payoff is not maximal for any of the players.